ABSTRACT. Let A be a k-algebra graded by a finite group G, with A 1 the component for the identity element of G. We consider such a grading as a "coaction" by G, in that A is a k[G]*-module algebra. We then study the smash product A#k[G]*; it plays a role similar to that played by the skew group ring R * G in the case of group actions. and enables us to obtain results relating the modules over A, A I' and A#k[G]*. After giving algebraic versions of the Duality Theorems for Actions and Coactions (results coming from von Neumann algebras), we apply them to study the prime ideals of A and A I' In particular we generalize Lorenz and Passman's theorem on incomparability of primes in crossed products. We also answer a question of Bergman on graded Jacobson radicals.

This paper develops a theory of crossed products and inner (weak) actions of arbitrary Hopf algebras on noncommutative algebras. The theory covers the usual examples of inner automorphisms and derivations, and in addition is general enough to include "inner" group gradings of algebras. We prove that if 7r : H-► H is a Hopf algebra epimorphism which is split as a coalgebra map, then H is algebra isomorphic to A #" H, a crossed product of H with the left Hopf kernel A of it. Given any crossed product A #CT H with H (weakly) inner on A, then A #CT if is isomorphic to a twisted product AT[H] with trivial action. Finally, if H is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of A implies that of A #" H; in particular this is true if the (weak) action of H is inner.

No abstract

Let A be a k-algebra graded by a finite group G, with A 1 the component for the identity element of G. We consider such a grading as a "coaction" by G, in that A is a k[G]*-module algebra. We then study the smash product A#k[G]*; it plays a role similar to that played by the skew group ring R * G in the case of group actions. and enables us to obtain results relating the modules over A, A I' and A#k[G]*. After giving algebraic versions of the Duality Theorems for Actions and Coactions (results coming from von Neumann algebras), we apply them to study the prime ideals of A and A I' In particular we generalize Lorenz and Passman's theorem on incomparability of primes in crossed products. We also answer a question of Bergman on graded Jacobson radicals.

No abstract